General Recipe Trouble Shooting

Attemped to make Fettuccine Alfredo for the first time.  I used heavy cream, butter, shredded Romano and Parmigiano-Reggiano.  I melted the butter over Med/Low, then slowly added the cheese a bit at a time.  I brought the whole sauce to a slow simmer for a few minutes then took it off the fire.  It tasted good.  However, it was slightly gritty and I'm trying to figure out why.  Did I bring the temp too high?  Or maybe not high enough?  How can I make it smooth?